# Copyright (C) 2010, Jesper Friis
# (see accompanying license files for details).
"""Definition of the Spacegroup class.
This module only depends on NumPy and the space group database.
"""
import os
import warnings
import numpy as np
__all__ = ['Spacegroup']
class SpacegroupError(Exception):
"""Base exception for the spacegroup module."""
pass
class SpacegroupNotFoundError(SpacegroupError):
"""Raised when given space group cannot be found in data base."""
pass
class SpacegroupValueError(SpacegroupError):
"""Raised when arguments have invalid value."""
pass
[docs]class Spacegroup(object):
"""A space group class.
The instances of Spacegroup describes the symmetry operations for
the given space group.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>>
>>> sg = Spacegroup(225)
>>> print 'Space group', sg.no, sg.symbol
Space group 225 F m -3 m
>>> sg.scaled_primitive_cell
array([[ 0. , 0.5, 0.5],
[ 0.5, 0. , 0.5],
[ 0.5, 0.5, 0. ]])
>>> sites, kinds = sg.equivalent_sites([[0,0,0]])
>>> sites
array([[ 0. , 0. , 0. ],
[ 0. , 0.5, 0.5],
[ 0.5, 0. , 0.5],
[ 0.5, 0.5, 0. ]])
"""
no = property(
lambda self: self._no,
doc='Space group number in International Tables of Crystallography.')
symbol = property(
lambda self: self._symbol,
doc='Hermann-Mauguin (or international) symbol for the space group.')
setting = property(
lambda self: self._setting,
doc='Space group setting. Either one or two.')
lattice = property(
lambda self: self._symbol[0],
doc="""Lattice type:
P primitive
I body centering, h+k+l=2n
F face centering, h,k,l all odd or even
A,B,C single face centering, k+l=2n, h+l=2n, h+k=2n
R rhombohedral centering, -h+k+l=3n (obverse); h-k+l=3n (reverse)
""")
centrosymmetric = property(
lambda self: self._centrosymmetric,
doc='Whether a center of symmetry exists.')
scaled_primitive_cell = property(
lambda self: self._scaled_primitive_cell,
doc='Primitive cell in scaled coordinates as a matrix with the '
'primitive vectors along the rows.')
reciprocal_cell = property(
lambda self: self._reciprocal_cell,
doc='Tree Miller indices that span all kinematically non-forbidden '
'reflections as a matrix with the Miller indices along the rows.')
nsubtrans = property(
lambda self: len(self._subtrans),
doc='Number of cell-subtranslation vectors.')
def _get_nsymop(self):
"""Returns total number of symmetry operations."""
if self.centrosymmetric:
return 2 * len(self._rotations) * len(self._subtrans)
else:
return len(self._rotations) * len(self._subtrans)
nsymop = property(_get_nsymop, doc='Total number of symmetry operations.')
subtrans = property(
lambda self: self._subtrans,
doc='Translations vectors belonging to cell-sub-translations.')
rotations = property(
lambda self: self._rotations,
doc='Symmetry rotation matrices. The invertions are not included '
'for centrosymmetrical crystals.')
translations = property(
lambda self: self._translations,
doc='Symmetry translations. The invertions are not included '
'for centrosymmetrical crystals.')
def __init__(self, spacegroup, setting=1, datafile=None):
"""Returns a new Spacegroup instance.
Parameters:
spacegroup : int | string | Spacegroup instance
The space group number in International Tables of
Crystallography or its Hermann-Mauguin symbol. E.g.
spacegroup=225 and spacegroup='F m -3 m' are equivalent.
setting : 1 | 2
Some space groups have more than one setting. `setting`
determines Which of these should be used.
datafile : None | string
Path to database file. If `None`, the the default database
will be used.
"""
if isinstance(spacegroup, Spacegroup):
for k, v in spacegroup.__dict__.iteritems():
setattr(self, k, v)
return
if not datafile:
datafile = get_datafile()
f = open(datafile, 'r')
try:
_read_datafile(self, spacegroup, setting, f)
finally:
f.close()
def __repr__(self):
return 'Spacegroup(%d, setting=%d)' % (self.no, self.setting)
[docs] def __str__(self):
"""Return a string representation of the space group data in
the same format as found the database."""
retval = []
# no, symbol
retval.append('%-3d %s\n' % (self.no, self.symbol))
# setting
retval.append(' setting %d\n' % (self.setting))
# centrosymmetric
retval.append(' centrosymmetric %d\n' % (self.centrosymmetric))
# primitive vectors
retval.append(' primitive vectors\n')
for i in range(3):
retval.append(' ')
for j in range(3):
retval.append(' %13.10f' % (self.scaled_primitive_cell[i, j]))
retval.append('\n')
# primitive reciprocal vectors
retval.append(' reciprocal vectors\n')
for i in range(3):
retval.append(' ')
for j in range(3):
retval.append(' %3d' % (self.reciprocal_cell[i, j]))
retval.append('\n')
# sublattice
retval.append(' %d subtranslations\n' % self.nsubtrans)
for i in range(self.nsubtrans):
retval.append(' ')
for j in range(3):
retval.append(' %13.10f' % (self.subtrans[i, j]))
retval.append('\n')
# symmetry operations
nrot = len(self.rotations)
retval.append(' %d symmetry operations (rot+trans)\n' % nrot)
for i in range(nrot):
retval.append(' ')
for j in range(3):
retval.append(' ')
for k in range(3):
retval.append(' %2d' % (self.rotations[i, j, k]))
retval.append(' ')
for j in range(3):
retval.append(' %13.10f' % self.translations[i, j])
retval.append('\n')
retval.append('\n')
return ''.join(retval)
[docs] def __eq__(self, other):
"""Chech whether *self* and *other* refer to the same
spacegroup number and setting."""
if not isinstance(other, Spacegroup):
other = Spacegroup(other)
return self.no == other.no and self.setting == other.setting
def __index__(self):
return self.no
[docs] def get_symop(self):
"""Returns all symmetry operations (including inversions and
subtranslations) as a sequence of (rotation, translation)
tuples."""
symop = []
parities = [1]
if self.centrosymmetric:
parities.append(-1)
for parity in parities:
for subtrans in self.subtrans:
for rot, trans in zip(self.rotations, self.translations):
newtrans = np.mod(trans + subtrans, 1)
symop.append((parity*rot, newtrans))
return symop
[docs] def get_op(self):
"""Returns all symmetry operations (including inversions and
subtranslations), but unlike get_symop(), they are returned as
two ndarrays."""
if self.centrosymmetric:
rot = np.tile(np.vstack((self.rotations, -self.rotations)),
(self.nsubtrans, 1, 1))
trans = np.repeat(self.subtrans, 2*len(self.rotations), axis=0)
else:
rot = np.tile(self.rotations, (self.nsubtrans, 1, 1))
trans = np.repeat(self.subtrans, len(self.rotations), axis=0)
return rot, trans
[docs] def get_rotations(self):
"""Return all rotations, including inversions for
centrosymmetric crystals."""
if self.centrosymmetric:
return np.vstack((self.rotations, -self.rotations))
else:
return self.rotations
[docs] def equivalent_reflections(self, hkl):
"""Return all equivalent reflections to the list of Miller indices
in hkl.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>> sg = Spacegroup(225) # fcc
>>> sg.equivalent_reflections([[0, 0, 2]])
array([[ 0, 0, -2],
[ 0, -2, 0],
[-2, 0, 0],
[ 2, 0, 0],
[ 0, 2, 0],
[ 0, 0, 2]])
"""
hkl = np.array(hkl, dtype='int', ndmin=2)
rot = self.get_rotations()
n, nrot = len(hkl), len(rot)
R = rot.transpose(0, 2, 1).reshape((3*nrot, 3)).T
refl = np.dot(hkl, R).reshape((n*nrot, 3))
ind = np.lexsort(refl.T)
refl = refl[ind]
diff = np.diff(refl, axis=0)
mask = np.any(diff, axis=1)
return np.vstack((refl[mask], refl[-1,:]))
[docs] def symmetry_normalised_reflections(self, hkl):
"""Returns an array of same size as *hkl*, containing the
corresponding symmetry-equivalent reflections of lowest
indices.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>> sg = Spacegroup(225) # fcc
>>> sg.symmetry_normalised_reflections([[2, 0, 0], [0, 2, 0]])
array([[ 0, 0, -2],
[ 0, 0, -2]])
"""
hkl = np.array(hkl, dtype=int, ndmin=2)
normalised = np.empty(hkl.shape, int)
R = self.get_rotations().transpose(0, 2, 1)
for i, g in enumerate(hkl):
gsym = np.dot(R, g)
j = np.lexsort(gsym.T)[0]
normalised[i,:] = gsym[j]
return normalised
[docs] def unique_reflections(self, hkl):
"""Returns a subset *hkl* containing only the symmetry-unique
reflections.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>> sg = Spacegroup(225) # fcc
>>> sg.unique_reflections([[ 2, 0, 0],
... [ 0, -2, 0],
... [ 2, 2, 0],
... [ 0, -2, -2]])
array([[2, 0, 0],
[2, 2, 0]])
"""
hkl = np.array(hkl, dtype=int, ndmin=2)
hklnorm = self.symmetry_normalised_reflections(hkl)
perm = np.lexsort(hklnorm.T)
iperm = perm.argsort()
xmask = np.abs(np.diff(hklnorm[perm], axis=0)).any(axis=1)
mask = np.concatenate(([True], xmask))
imask = mask[iperm]
return hkl[imask]
[docs] def equivalent_sites(self, scaled_positions, ondublicates='error',
symprec=1e-3):
"""Returns the scaled positions and all their equivalent sites.
Parameters:
scaled_positions: list | array
List of non-equivalent sites given in unit cell coordinates.
ondublicates : 'keep' | 'replace' | 'warn' | 'error'
Action if `scaled_positions` contain symmetry-equivalent
positions:
'keep'
ignore additional symmetry-equivalent positions
'replace'
replace
'warn'
like 'keep', but issue an UserWarning
'error'
raises a SpacegroupValueError
symprec: float
Minimum "distance" betweed two sites in scaled coordinates
before they are counted as the same site.
Returns:
sites: array
A NumPy array of equivalent sites.
kinds: list
A list of integer indices specifying which input site is
equivalent to the corresponding returned site.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>> sg = Spacegroup(225) # fcc
>>> sites, kinds = sg.equivalent_sites([[0, 0, 0], [0.5, 0.0, 0.0]])
>>> sites
array([[ 0. , 0. , 0. ],
[ 0. , 0.5, 0.5],
[ 0.5, 0. , 0.5],
[ 0.5, 0.5, 0. ],
[ 0.5, 0. , 0. ],
[ 0. , 0.5, 0. ],
[ 0. , 0. , 0.5],
[ 0.5, 0.5, 0.5]])
>>> kinds
[0, 0, 0, 0, 1, 1, 1, 1]
"""
kinds = []
sites = []
symprec2 = symprec**2
scaled = np.array(scaled_positions, ndmin=2)
for kind, pos in enumerate(scaled):
for rot, trans in self.get_symop():
site = np.mod(np.dot(rot, pos) + trans, 1.)
if not sites:
sites.append(site)
kinds.append(kind)
continue
t = site - sites
mask = np.sum(t*t, 1) < symprec2
if np.any(mask):
ind = np.argwhere(mask)[0][0]
if kinds[ind] == kind:
pass
elif ondublicates == 'keep':
pass
elif ondublicates == 'replace':
kinds[ind] = kind
elif ondublicates == 'warn':
warnings.warn('scaled_positions %d and %d '
'are equivalent'%(kinds[ind], kind))
elif ondublicates == 'error':
raise SpacegroupValueError(
'scaled_positions %d and %d are equivalent'%(
kinds[ind], kind))
else:
raise SpacegroupValueError(
'Argument "ondublicates" must be one of: '
'"keep", "replace", "warn" or "error".')
else:
sites.append(site)
kinds.append(kind)
return np.array(sites), kinds
[docs] def symmetry_normalised_sites(self, scaled_positions):
"""Returns an array of same size as *scaled_positions*,
containing the corresponding symmetry-equivalent sites within
the unit cell of lowest indices.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>> sg = Spacegroup(225) # fcc
>>> sg.symmetry_normalised_sites([[0.0, 0.5, 0.5], [1.0, 1.0, 0.0]])
array([[ 0., 0., 0.],
[ 0., 0., 0.]])
"""
scaled = np.array(scaled_positions, ndmin=2)
normalised = np.empty(scaled.shape, np.float)
rot, trans = self.get_op()
for i, pos in enumerate(scaled):
sympos = np.dot(rot, pos) + trans
# Must be done twice, see the scaled_positions.py test
sympos %= 1.0
sympos %= 1.0
j = np.lexsort(sympos.T)[0]
normalised[i,:] = sympos[j]
return normalised
[docs] def unique_sites(self, scaled_positions, symprec=1e-3, output_mask=False):
"""Returns a subset of *scaled_positions* containing only the
symmetry-unique positions. If *output_mask* is True, a boolean
array masking the subset is also returned.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>> sg = Spacegroup(225) # fcc
>>> sg.unique_sites([[0.0, 0.0, 0.0],
... [0.5, 0.5, 0.0],
... [1.0, 0.0, 0.0],
... [0.5, 0.0, 0.0]])
array([[ 0. , 0. , 0. ],
[ 0.5, 0. , 0. ]])
"""
scaled = np.array(scaled_positions, ndmin=2)
symnorm = self.symmetry_normalised_sites(scaled)
perm = np.lexsort(symnorm.T)
iperm = perm.argsort()
xmask = np.abs(np.diff(symnorm[perm], axis=0)).max(axis=1) > symprec
mask = np.concatenate(([True], xmask))
imask = mask[iperm]
if output_mask:
return scaled[imask], imask
else:
return scaled[imask]
[docs] def tag_sites(self, scaled_positions, symprec=1e-3):
"""Returns an integer array of the same length as *scaled_positions*,
tagging all equivalent atoms with the same index.
Example:
>>> from ase.lattice.spacegroup import Spacegroup
>>> sg = Spacegroup(225) # fcc
>>> sg.tag_sites([[0.0, 0.0, 0.0],
... [0.5, 0.5, 0.0],
... [1.0, 0.0, 0.0],
... [0.5, 0.0, 0.0]])
array([0, 0, 0, 1])
"""
scaled = np.array(scaled_positions, ndmin=2)
scaled %= 1.0
scaled %= 1.0
tags = -np.ones((len(scaled), ), dtype=int)
mask = np.ones((len(scaled), ), dtype=np.bool)
rot, trans = self.get_op()
i = 0
while mask.any():
pos = scaled[mask][0]
sympos = np.dot(rot, pos) + trans
# Must be done twice, see the scaled_positions.py test
sympos %= 1.0
sympos %= 1.0
m = ~np.all(np.any(np.abs(scaled[np.newaxis,:,:] -
sympos[:,np.newaxis,:]) > symprec,
axis=2), axis=0)
assert not np.any((~mask) & m)
tags[m] = i
mask &= ~m
i += 1
return tags
def get_datafile():
"""Return default path to datafile."""
return os.path.join(os.path.dirname(__file__), 'spacegroup.dat')
def format_symbol(symbol):
"""Returns well formatted Hermann-Mauguin symbol as extected by
the database, by correcting the case and adding missing or
removing dublicated spaces."""
fixed = []
s = symbol.strip()
s = s[0].upper() + s[1:].lower()
for c in s:
if c.isalpha():
fixed.append(' ' + c + ' ')
elif c.isspace():
fixed.append(' ')
elif c.isdigit():
fixed.append(c)
elif c == '-':
fixed.append(' ' + c)
elif c == '/':
fixed.append(' ' + c)
s = ''.join(fixed).strip()
return ' '.join(s.split())
#-----------------------------------------------------------------
# Functions for parsing the database. They are moved outside the
# Spacegroup class in order to make it easier to later implement
# caching to avoid reading the database each time a new Spacegroup
# instance is created.
#-----------------------------------------------------------------
def _skip_to_blank(f, spacegroup, setting):
"""Read lines from f until a blank line is encountered."""
while True:
line = f.readline()
if not line:
raise SpacegroupNotFoundError(
'invalid spacegroup %s, setting %i not found in data base' %
( spacegroup, setting ) )
if not line.strip():
break
def _skip_to_nonblank(f, spacegroup, setting):
"""Read lines from f until a nonblank line not starting with a
hash (#) is encountered and returns this and the next line."""
while True:
line1 = f.readline()
if not line1:
raise SpacegroupNotFoundError(
'invalid spacegroup %s, setting %i not found in data base' %
( spacegroup, setting ) )
line1.strip()
if line1 and not line1.startswith('#'):
line2 = f.readline()
break
return line1, line2
def _read_datafile_entry(spg, no, symbol, setting, f):
"""Read space group data from f to spg."""
spg._no = no
spg._symbol = symbol.strip()
spg._setting = setting
spg._centrosymmetric = bool(int(f.readline().split()[1]))
# primitive vectors
f.readline()
spg._scaled_primitive_cell = np.array([
list(map(float, f.readline().split()))
for i in range(3)],
dtype='float')
# primitive reciprocal vectors
f.readline()
spg._reciprocal_cell = np.array([list(map(int, f.readline().split()))
for i in range(3)],
dtype=np.int)
# subtranslations
spg._nsubtrans = int(f.readline().split()[0])
spg._subtrans = np.array([list(map(float, f.readline().split()))
for i in range(spg._nsubtrans)],
dtype=np.float)
# symmetry operations
nsym = int(f.readline().split()[0])
symop = np.array([list(map(float, f.readline().split())) for i in range(nsym)],
dtype=np.float)
spg._nsymop = nsym
spg._rotations = np.array(symop[:,:9].reshape((nsym,3,3)), dtype=np.int)
spg._translations = symop[:,9:]
def _read_datafile(spg, spacegroup, setting, f):
if isinstance(spacegroup, int):
pass
elif isinstance(spacegroup, str):
#spacegroup = ' '.join(spacegroup.strip().split())
spacegroup = format_symbol(spacegroup)
else:
raise SpacegroupValueError('`spacegroup` must be of type int or str')
while True:
line1, line2 = _skip_to_nonblank(f, spacegroup, setting)
_no,_symbol = line1.strip().split(None, 1)
_symbol = format_symbol(_symbol)
_setting = int(line2.strip().split()[1])
_no = int(_no)
if ((isinstance(spacegroup, int) and _no == spacegroup) or
(isinstance(spacegroup, str) and
_symbol == spacegroup)) and _setting == setting:
_read_datafile_entry(spg, _no, _symbol, _setting, f)
break
else:
_skip_to_blank(f, spacegroup, setting)
def parse_sitesym(symlist, sep=','):
"""Parses a sequence of site symmetries in the form used by
International Tables and returns corresponding rotation and
translation arrays.
Example:
>>> symlist = [
... 'x,y,z',
... '-y+1/2,x+1/2,z',
... '-y,-x,-z',
... ]
>>> rot, trans = parse_sitesym(symlist)
>>> rot
array([[[ 1, 0, 0],
[ 0, 1, 0],
[ 0, 0, 1]],
<BLANKLINE>
[[ 0, -1, 0],
[ 1, 0, 0],
[ 0, 0, 1]],
<BLANKLINE>
[[ 0, -1, 0],
[-1, 0, 0],
[ 0, 0, -1]]])
>>> trans
array([[ 0. , 0. , 0. ],
[ 0.5, 0.5, 0. ],
[ 0. , 0. , 0. ]])
"""
nsym = len(symlist)
rot = np.zeros((nsym, 3, 3), dtype='int')
trans = np.zeros((nsym, 3))
for i, sym in enumerate(symlist):
for j, s in enumerate (sym.split(sep)):
s = s.lower().strip()
while s:
sign = 1
if s[0] in '+-':
if s[0] == '-':
sign = -1
s = s[1:]
if s[0] in 'xyz':
k = ord(s[0]) - ord('x')
rot[i, j, k] = sign
s = s[1:]
elif s[0].isdigit() or s[0] == '.':
n = 0
while n < len(s) and (s[n].isdigit() or s[n] in '/.'):
n += 1
t = s[:n]
s = s[n:]
if '/' in t:
q, r = t.split('/')
trans[i,j] = float(q)/float(r)
else:
trans[i,j] = float(t)
else:
raise SpacegroupValueError(
'Error parsing %r. Invalid site symmetry: %s' %
(s, sym))
return rot, trans
def spacegroup_from_data(no=None, symbol=None, setting=1,
centrosymmetric=None, scaled_primitive_cell=None,
reciprocal_cell=None, subtrans=None, sitesym=None,
rotations=None, translations=None, datafile=None):
"""Manually create a new space group instance. This might be
usefull when reading crystal data with its own spacegroup
definitions."""
if no is not None:
spg = Spacegroup(no, setting, datafile)
elif symbol is not None:
spg = Spacegroup(symbol, setting, datafile)
else:
raise SpacegroupValueError('either *no* or *symbol* must be given')
have_sym = False
if centrosymmetric is not None:
spg._centrosymmetric = bool(centrosymmetric)
if scaled_primitive_cell is not None:
spg._scaled_primitive_cell = np.array(scaled_primitive_cell)
if reciprocal_cell is not None:
spg._reciprocal_cell = np.array(reciprocal_cell)
if subtrans is not None:
spg._subtrans = np.atleast_2d(subtrans)
spg._nsubtrans = spg._subtrans.shape[0]
if sitesym is not None:
spg._rotations, spg._translations = parse_sitesym(sitesym)
have_sym = True
if rotations is not None:
spg._rotations = np.atleast_3d(rotations)
have_sym = True
if translations is not None:
spg._translations = np.atleast_2d(translations)
have_sym = True
if have_sym:
if spg._rotations.shape[0] != spg._translations.shape[0]:
raise SpacegroupValueError('inconsistent number of rotations and '
'translations')
spg._nsymop = spg._rotations.shape[0]
return spg
#-----------------------------------------------------------------
# Self test
if __name__ == '__main__':
# Import spacegroup in order to ensure that __file__ is defined
# such that the data base can be found.
import spacegroup
import doctest
print('doctest: ', doctest.testmod())